Dimensions of learning as identity: A demonstration fromculturally-based lessons in Grade 9 mathematics

*Madusise, Sylvia & Mwakapenda, WillyDepartment of Mathematics, Science and Technology Education,Tshwane University of Technology, Soshanguve North Campus

MadusiseS@tut.ac.za, MwakapendaWWJ@tut.ac.za

This qualitative study examined the impact of culturally-relevant mathematics lessons on learners’ mathematicalidentity. Three Grade 9 mathematics teachers and theirlearners from one rural middle school in South Africa’s NorthWest Province participated in the study. Through mathematisingculturally-based activities, the research team indigenised(i.e. adapted to local culture) two Grade 9 mathematicstopics. A teaching and learning unit on the indigenised topicswas designed and implemented in five Grade 9 classes at thesame school. In the discussion in this paper, we considermathematics learning as a process of developing a mathematicalidentity. We address how Grade 9 learners’ practices inmathematics classroom communities shape learners’ perspectivesof themselves. In this paper we view identity in two ways: howindividuals know and name themselves, and how individuals arerecognised and looked upon by others. We argue thatculturally-relevant pedagogy can facilitate the development oflearners’ mathematical identities by rationally usinglearners’ cultures to serve as a bridge between the learnerand the subject. Learners’ mathematical identities weredetermined through analysing their narratives of mathematicsclassroom practice. We argue that the three interrelateddimensions: Competence, Performance and Recognition can beused to describe learning as identity. The paper analyseslearners’ narratives of mathematics classroom practices inGrade 9 to demonstrate these dimensions of identity in theactivity of mathematics learning. Key Words: Culturally-relevant pedagogy, identity, mathematicsclassroom community

Background to the study

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The cultural placement of an educational system is probablythe most relevant fact in modern development of education(D’Ambrosio, 1979). Many researchers have agreed that teachingmust be related to its cultural and geographical context(Bishop, 1988; Kroma, 1996; Ascher, 1994; Gerdes, 1999;Mosimege, 2003; Madusise, 2010). There is consensus amongstthese researchers that mathematics is being perceived as dry,uninteresting and irrelevant. Familiar subject matter andexperiences that could be used to lay the foundations of thediscipline, arouse learners’ interests and challenge theirintellect early in life have been largely neglected (Kroma,1996)

South Africa has embarked upon a curriculum that strives toenable all learners to achieve to their maximum potential(Revised National Curriculum Policy, 2002). Curriculumoutcomes encourage learner-centred and activity-basedapproaches. Policy statements for Grades R-9 Mathematicsenvisage learners who will “be culturally and aestheticallysensitive across a range of social contexts” (Department ofEducation, 2002: 2). Interestingly, some assessment standardsexpect learners to be able to solve problems in contexts thatmay be used to build awareness of social, cultural andenvironmental issues. The National Curriculum Statement (NCS)challenges educators to find new and innovative ways to reachstudents from diverse cultures in their mathematicsclassrooms. Valuing indigenous knowledge systems is one of theprinciples upon which the NCS is based.Various theorists have outlined pedagogical strategies whichincorporate students’ cultural backgrounds and priorexperiences. Ladson- Billings (1994) asserts that culturallyrelevant teaching is designed not only to fit the schoolculture to the students’ culture but also to use students’cultures as a basis for helping students understand themselvesand conceptualise knowledge. Culturally relevant pedagogy hasbeen defined as a means to use students’ cultures to bridgecultural knowledge and school knowledge (Boutte and Hill,2006), to validate students’ life experiences by utilisingtheir cultures, histories as teaching resources (Boyle-Baise,2005), to recognize students’ home cultures, promotecollaboration among peers, and connect home life with school

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experiences (Neuman, 1999). It would appear the proponents ofthis pedagogy generally contend that culturally responsiveteaching acknowledges the legitimacy of the cultural heritagesof different ethnic groups as legacies that affect students’dispositions, attitudes and approaches to learning. Studiesbased on the concept of cultural differences make anassumption that students coming from culturally diversebackgrounds will achieve academic excellence if classroominstruction is conducted in a manner responsive to thestudents’ home culture (de Beer, 2010).Many mathematicians, mathematics teachers and students possess“only a limited understanding of what and how [cultural]values are being transmitted” through the discipline (Bishop,2001, p.234). Culturally relevant mathematics lessons workagainst this ignorance by reversing the trend in traditionalmathematics curricula to divorce mathematics from its culturalroots (Troutman &McCoy, 2008). Ladson-Billings (1995) documented the success of innovativelessons that appeal to diverse cultures in improving students’attitudes towards classroom subject matter. Teachers whoparticipated in her study developed lessons that incorporatedthe knowledge students gained from their lives outside ofclass and demonstrating the value of students’ home culturesand languages. By so doing the participating teacherspositively influenced student test scores, engagement in theclassroom community, and overall attitude towards school andlearning (Ladson-Billings, 1995). Lloyd (2001) conducted a five-year study aimed at showingteachers alternative classroom practices to better meet theneeds of students. The study focussed on encouraging schoolsto promote the learning by all students through lessons thatshowed connections between mathematics and students’ lives.The most successful educators were those who fully embracedinnovative and culturally relevant lessons. The study reportedin this paper aimed to connect the teaching and learning ofmathematics to cultural contexts, checking on the potential ofculturally-based mathematics lessons for influencing learners’mathematical identities.

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Theoretical framework: Identity development as a lens formathematics learningMany researchers and theorists have considered how mathematicsis learned, each foregrounding different aspects of thelearning processes and viewing it from a range of theoreticalframeworks. Learning mathematics can be viewed as buildingskills, using algorithms and following certain procedures.Another view focusses on students’ construction or acquisitionof mathematical concepts (Anderson, 2007). This study focusseson the view that learning mathematics in schools involvesbecoming a ‘certain type’ of person with respect to thepractices of a community. In this view, learning occursthrough “social participation” (Wenger, 1998, p.4): learning“changes who we are by changing our ability to participate, tobelong and to negotiate meaning” (Wenger, 1998, p.226).According to Wenger all learning eventually gains significancein the kind of person we become.This paper addresses how Grade 9 learners’ practices withinmathematics classroom communities shape learners’ sense ofthemselves - their identities.

Learning transforms our identities: it transforms our abilityto participate in the world by changing all at once who we are,our practices, our communities, learning is a matter ofengagement: it depends on opportunities to contribute activelyto the practices of communities that we value (Wenger, 1998,p. 227).

Therefore learning mathematics also involves the developmentof each learner’s identity as a member of the mathematicsclassroom community. In this paper we view identity as “howindividuals know and name themselves and how an individual isrecognised and looked upon by others” (Grootenboer et al,2006, p.612). Identity refers to the way we define ourselvesand how others define us (Sfard & Prusak, 2005; Wenger, 1998).Thus identity is socially constituted, that is, one isrecognised by self and others as a kind of person because ofthe interactions one has with others. Putman and Borko (2000) stated that, “how a person learns aparticular set of knowledge and skills, and the situation inwhich a person learns becomes a fundamental part of what islearned” (p.4). With this in mind, the impact of culturally-based mathematics lessons was determined through analysing

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learners’ narratives. Identity was then seen as a set ofreifying, significant and endorsable stories about a person(Sfard & Prusak, 2005). Learning events and forms ofparticipation are thus defined by the current engagement theyafford. Problems of disengagement and non-participation inmathematics have tenaciously persisted for many years (Bishop,2001). It is hoped that an investigation of mathematicallearning as identity development through engaging inculturally relevant mathematics lessons may offer some newinsights into how these issues can be ameliorated.

A possible mathematics identity modelIn this section, we describe our mathematics identity modeldeveloped to provide analytical direction for our dataanalysis and interpretation. The model was adapted fromCarlone and Johnson (2007)’s Initial Science Identity Modelwhich is premised on both practical and theoretical sources ofdata. In our model a person who has a strong mathematicsidentity is described as someone who is competent; whodemonstrates meaningful knowledge and understanding ofmathematics content, who can perform for others hercompetences with mathematical practices/activities, andfinally, someone who can recognise herself / himself, and getsrecognised by others as a mathematics learner. These aspectsof mathematics identity are captured in the followinginterrelated dimensions: competence, performance and recognition.

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Performance Recognit

ion

Social performances of relevantRecognizing oneself and getting recognized by mathematical practicesothers as a “mathematics learner”

Generating scenarios:. images of ourselves

visualizations. images of possibilities

simulations. images of the past and the future

unique experiences

Acquisition of knowledge and understanding of mathematics content reflected in shared histories of learning. [Community of practice can give participants access to competence andacquisition of knowledge]

Figure 1: Model of mathematics identityAny one of the three dimensions illustrated in the diagram inFigure 1 can be used to describe mathematical identity. However theintersection of either two or all the three sets can representan even stronger mathematical identity.

Research questionThis paper addresses the following central research question:What is the potential of culturally-based lessons in Grade 9 mathematics forinfluencing learners’ mathematical identities?

Methodology

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Competence

Mathematics Identity

To address the above question, data sources included learners’pre and post questionnaire, teacher and learners’ interviewtranscripts, learners’ lesson journals, lesson observations,teachers’ reflective forms and transcripts from reflectivemeetings. These multiple data sources (Merriam, 1998; Yin,2003) served as corroborating evidence to enrich the pictureof teaching practices presented in the study and the storieslearners tell about their engagement in the mathematicslearning communities. The multiple sources of data providedconvergent lines of evidence to enhance credibility ofassertions (Yin, 2003).Quotations from learners’ questionnaire responses andinterview transcripts were used as narrative stories told bylearners evaluating their participation. Excerpts fromteachers’ interview transcripts and reflective meetings wereused to find out how teachers recognised learners as a part ofthe mathematics learning community (Boaler, 2000; Wenger,1998). Therefore the excerpts were used to determine learners’mathematical identity, using the identity-as-narrativeconstruct (Sfard & Prusak, 2005). Sfard and Prusak clarifythat the identity-as-narrative construct assumes identity tobe “human-made” rather than “God-given” (p.17). They assertthat narrative is not a window onto identity, rather narrativeis identity (p.14). Thus they suggest, identity is adiscursive construct where identities are considered as“discursive counterparts to lived experiences” (p.17) limitingtheir definition to linguistic signs that index or represent aperson’s lived experience. In agreement with this definitionwe used linguistic data from learners’ experiences to count asadmissible evidence in constructing an account of identity.In our model we considered identity as one of those self-evident notions that, whether reflectively or instinctively,arise from one’s first-hand experience. This is linked to thework of Lave and Wenger. According to Lave and Wenger (1991,p. 53) “learning… implies becoming a different person…learning involves the construction of identity”. Therefore,“being a kind of person” remains the centrepiece of thedefinition of identity. The motif of a “person’s ownnarrativization” recurs in the description proposed by Hollandet al. (1998):

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People tell others who they are, but even more importantly,they tell themselves and they try to act as though they are whothey say they are. These self-understandings, especially thosewith strong emotional resonance for teller, are what we referto as identities (p.3).

Samples and sampling proceduresThe sample in this study consisted of three mathematicsteachers from one middle rural school in the North WestProvince of South Africa and their Grade 9 learners. Purposiveand convenience sampling was used to select the research sites(Patton, 1990). Merriam (2009) identifies purposive samplingas one appropriate sampling strategy in case-study design.Merriam (2009) further adds that purposeful sampling is basedon the assumption that one wants to discover, understand, gainsight; therefore one needs to select a sample from which onecan learn the most. In this case, a cultural village1 wasidentified as the research site and mathematics teachers whoteach at a school very close to the selected cultural villagewere focused on. A cultural village was selected with thebelief that it is where the community’s indigenous knowledgeis preserved. The intention was to make the cultural village amathematics teaching resource centre. A school close to thecultural village was chosen with an assumption that itsmembers (including learners) are quite familiar with theactivities taking place at the cultural village.Grade 9 was chosen based on the argument that it is atransitional grade from GET to FET where students after Grade9 are to choose between Mathematics and Mathematical Literacy.At Grade 9 students are learning mathematics which combinesaspects of both Mathematics and Mathematical Literacy. AtGrade 9 learners have more experience with mathematics thanlearners at earlier grades. Doing the study at FET level wouldhave limited the number of participating learners as somelearners might have perceived it as being linked toMathematical Literacy and would therefore withdraw sinceeveryday examples are usually associated with MathematicalLiteracy.

Intervention teaching context

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1 A cultural village is a tourist establishment where tourists can view aspects such as the homestead, traditional clothing, food and food-related practices, history and societal structures as well as song and dance

Two Grade 9 topics were taught using culturally-basedactivities in five Grade 9 classes. The Setswana step dance, acultural dance practised at a cultural village near theschool, was used as a context for teaching number patterns. Agroup of Grade 9 learners1 demonstrated the dance and a numberpattern was observed involving the number of dancers and thenumber of foot-steps (see table below). The second row showsthe total number of steps made by all the dancers if eachdancer is making five steps.

Number of dancers 1 2 3 4 - - - - nNumber of foot-steps 5 10 15 20 - - - - nx5

Some tasks were formulated using the artefacts from thecultural village. Step dancing was again used to teachsubstitution using number of dancers as input and number ofsteps as output. In another topic, artefacts from the Ndebelepaintings and beadings, (see Figure 2 below) were used toteach properties of shapes and transformations.

Figure 2: Ndebele beadings

Three lessons were taught in each of the three classes (twofor Teacher A and one for Teacher C). All the 9 lessons weretaught by the researcher, while the class teachers were takingvideos. Four lessons were taught in each of Teacher B’s twoclasses. Three of these lessons in each of Teacher B’s classeswere taught by the researcher and one in each class by the1 These learners had previously participated in cultural dances at the Cultural Village near the school.

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class teacher (Teacher B). Altogether 17 lessons were taughtand video recorded. Reflective meetings were held after everylesson. Lessons were collaboratively planned by the researcherand the class teachers.

Nature of data and data analysisThe data collected in the study on which this paper ispremised included seventeen video-recorded culturally-basedlessons from five Grade 9 classes, learners’ responses frompre and post questionnaires, learners’ lesson journal entries,audio-recordings from learners’ group post- lesson interviews,audio-recordings of teachers’ pre and post interviews, notesfrom post-lesson reflective meetings with teachers andteachers’ lesson reflective forms. In this paper we focused onlearners’ narrative stories, on their perspectives ofculturally-based lessons, triangulating data from lessonjournal entries, post-lesson questionnaire responses and post-lesson interview transcripts. We also used teachers’narratives from the post-lesson reflective meetings and thepost-interview transcripts to check on how they recognisedlearners’ capabilities as mathematics learners.

Learners were asked to complete journal entries after everylesson, to reflect on their learning. Questions for the post-lesson questionnaire were constructed to check on learners’views about the importance of learning mathematics and theirperspectives on how the incorporation of culturally-basedactivities in mathematics lessons impacted on theirmathematics learning. Interviews were carried out to probelearners’ responses from journal entries and post-lessonquestionnaires as well as checking on learners’ perceivedlearning trajectories. In analysing the data, the narrativeswere accordingly coded using the themes; competence,performance and recognition – the dimensions of identity (seeFigure 1). The approach to analysis involved identifying the“words” learners used to represent their experiences and howsocial relationships and practices produced confidence andpride in the learning. When analysing learners’ responses,expressions like: I feel good, happy, excited, educated,impressed, cool, amazed etc., were categorised as indicating

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learner satisfaction. Narratives depicting the same theme weregrouped together into a vignette and commented on in detail.

Data presentation and discussionTo begin the analysis, completed questionnaires were givenidentity codes that represent the type of questionnaire, theclass of the learner and the number of the questionnaire. Thiswas also applied to the learners’ journals. For example aquestionnaire completed by a learner in Grade 9A wasidentified as Q1LA01, Q1 denoting first questionnaire and Q2denoting the final questionnaire, LA denoting learner fromGrade 9A, and 01 denoting the number of the questionnaire. ForJ1LA01, J1; J2; and J3 denote Lesson Journal 1, 2 and 3respectively and LA01 with the same representation as above.For GILA01, GI denote group interview.

In the following sections, vignettes are presented andcommented on to identify and demonstrate aspects of learningas identity.

Vignette 1: Factors which contribute to the difficulty ofmathematics This vignette contains extracts from learners’ responses onthe question: What factors do you think may contribute to the difficulty ofmathematics? Please explain. This question was designed to understandlearners’ participation as well as non-participation and ofinclusion as well as exclusion in the mathematics classroomcommunity. Learners were asked to answer the above questionprior the intervention teaching.

Q1LA02: In mathematics you work with numbers when in othersubjects you work with words, there are no notes or scenariosQ1LA08: Equations are difficult, you need to do all your stepscorrectly. If you miss some steps you get wrong answer. Q1B39: The way teachers teach. Other teachers explain veryfast, others do not know how to teach and they say we do notknow mathematics.Q1LC14: The difficult of mathematics is to work with algebraand equations. Things we do not understand give us stress orheadache.

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Q1LD24: The difficult factors for me are when you do thingswhich you do not talk about every day e.g. prime factorsbecause we do not talk about them every day.Q1LD06: Mathematics needs more time than other subjects.Q1LD41: Strict teachers contribute a lot to the difficult ofmathematics because sometimes you are afraid of them especiallywhen you do not get the required answer.Q1LE08: The difficult of mathematics is that you cannotcalculate the answer without following some steps. I just donot like going through each and every step without my ownshortcuts.Q1LE27: Unlike other subjects, it is difficult to readmathematics.

Comment Learners’ comments on the question were divided into sevencategories as follows: teaching and learning styles, natureof mathematics, insufficient resources, anxiety, procedures,language barrier, and insufficient time. However The wayteachers teach was cited as the major contributing factor tothe difficulty of mathematics. Eighty-one (81) out 157 (52%)learners cited factors related to this category. Reasons likemissing notes and scenarios (See Q1LA02; Q1LD24) were given.In the vignette above, Q1LA02 expressed dislike of learningmathematics without using related scenarios and Q1LD24emphasised that lack of connections to everyday life makesmathematics difficult. Learners who do not have theopportunity to connect with mathematics at a personal levelmay fail to see themselves as competent at learningmathematics (Boaler & Greeno, 2000). According to Q1LB39 thetreatments they get from their teachers contribute to theirnon-participation in the mathematics lessons. Learners who donot see themselves or are not recognised as contributors tothe mathematics classroom may not consider themselves to becapable mathematics learners. One way learners come to learn who they are relative tomathematics is through their engagement in the activities ofthe mathematics classroom. This idea of following laid downprocedures without own creativity does not go down well withsome learners (Q1LE08). When learners are able to developtheir own strategies and meanings for solving mathematics

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problems, rather than following laid down procedures, theylearn to view themselves as capable members of a communityengaged in mathematics learning (Boaler & Greeno, 2000).Therefore the types of mathematical tasks and teaching andlearning structures used in the classroom contributesignificantly to the learners’ mathematical identities. The nature of mathematics as a contributing factor to thedifficulty of mathematics was cited by 40 learners (~ 25%).Comments like, it is difficult to read mathematics (Q1LE27), in mathematicsyou work with numbers when in other subjects you work with words (Q1LA02), inmathematics there are some required answers (Q1LD41) were emphasised.Learners who are not the quickest to get the required answersmay see themselves as not capable of learning mathematics.Insufficient time (Q1LD06) and lack of resources likecalculators was also emphasised as contributing to thedifficulty of mathematics. For these learners the availabilityof learning resources is an important prerequisite formathematics learning. The use of English language inmathematics was also cited as another contributing factor tothe difficulty of mathematics since mathematics is a languagewhich is being taught in their second language (English).Therefore learners are struggling to understand these twolanguages simultaneously. Learners with problems in Englishlanguage may not see themselves as capable mathematicslearners.The above vignette highlights that, before the intervention,lack of good teaching strategies, good treatment by teachers,and connections to everyday life may have contributed tolearners’ weak mathematical identities (low capability ofdoing mathematics). Use of procedures and the nature ofmathematics may also contribute to a weak mathematicalidentity.

Vignette 2: Competence as an Identity DimensionIn this vignette the following narratives have been cited toshow learners’ comments on their acquisition of mathematicalknowledge and understanding of the mathematics content learntafter participating in culturally-based lessons.

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Q2LA05: I feel very amazed because I can see that othercultures they do mathematics in their own way which makes iteasy to know number patterns and transformations Q2LA28: I like the fact that I understand the whole topic(referring to number patterns), so I am excited when I getquestions based on the topic. I am proud of my work.J1LA21: Today I have learnt the following things in the numberpatterns; forming a number pattern from the Tswana dance,getting the position of the term in the number pattern, gettingthe general term and the rule connecting the terms.Q2LB02: I feel that number patterns are simple when we use theSetswana dance or any other dance from our culture. Because welisten and look, count their steps when they are dancing andget some mathematics.Q2LC02: I like using our culture because it made me understandmathsQ2LD26: I feel very impressed because I did not know numberpatterns but I now know them very well and I hope they can putthem in the examinations.Q2LC09: I feel like I can now handle anything in mathematics,mathematics is not all that difficult.Q2LE06: I liked these two topics because there is something inmy mind now that I have learnt about – it reminds me that inthe olden years in my culture, our elders used powerfulmathematics also.

CommentThe above quotations from learners overwhelmingly point to thesuccess of learners in their engagement and understanding theconcepts that were part of the culturally-based lessons. Outof 169 learners who responded to the questionnaire, 163 (96%)expressed satisfaction and 44% of the learners felt satisfiedwith the way they learnt the two indigenised topics becausethey gained a good understanding in these topics. Somelearners (e.g. Q2LD26) even wish to see questions on thecovered work in the examinations. This indicates a very highdegree of competence and confidence. This is furtherepitomised by Q2LC09 who feels competent to “…handle anythingin mathematics...” Q2LA28’s competence is indicated by his/hersentiments that s/he understands the whole topic (numberpatterns), excited to get questions based on the topic and

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proud of her/his work. Most learners expressed more or lessexplicitly their pleasure to learn mathematics and asatisfaction in completing a task. In the lesson journalsalmost every learner was able to express explicitly what s/helearnt during the intervention lessons (JILA21). This fluencyin mathematics talk demonstrates meaningful knowledge andunderstanding of the experienced mathematical content. Some of the learners in the study experienced the “aha”moments and joy of mathematical discovery, discovering a linkbetween mathematics in the classroom and the mathematics usedin the cultural activities. Their expressions (Q2LA05)revealed surprise in this gain of knowledge. Grootenboer andZevenbergen (2007) asserted that such personal responses,describing mathematics in terms of “wonder”, “beauty” and“delight” provide motivation in terms of continued engagementand fuel a passion for the discipline of mathematics. On theother hand, statements like those expressed by Q2LA05 andQ2LB02 suggest that, for them, the culturally relevant lessonsdid address Ladson-Billings’ (1995, p. 160) requirements forthis kind of teaching where learners are expected to“experience academic success, develop and/or maintain culturalcompetence”. Interestingly, knowing that even their forefathers were also doing what they termed powerful mathematics(Q2LE06) may spur them to acquire a strong mathematicalidentity. Their admission that they did not know thepossibility that some mathematical concepts might have evolvedin their cultures, not knowing that people in their cultureswere also doing mathematics in their cultural activities,prior to the lessons, suggests that the lessons encouraged thedevelopment of their “cultural competence” as mathematicslearners.

However, six out of 169 learners who responded to the post-lesson questionnaire expressed that they were unsatisfied withthe way they learnt the indigenised topics. Such learners hadthis to say:

Q2LA32: I was feeling so boring because I do not lovemathematics.Q2LB29: I feel nothing because I don’t like mathematics.

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For these learners, any innovative method introduced may nothave had a positive impact on them because they do not likemathematics. They attend mathematics lessons becausemathematics is a compulsory subject. Given a choice they woulddrop the subject.Vignette 3: Performance as an Identity dimensionBelow are some narratives from learners which indicate howthey valued their social performances in the culturally-basedlessons which were referred to as participation in a projectduring group interviews.

JILD01: I liked sharing my ideas with my teachers and otherlearners.J3LE12: I really liked the introduction using cultural songsand dance.Q2LE17: It was interesting. I was learning something now.Q2LE29: I learnt number patterns by the things that I couldunderstand and doing number patterns is very easy, you can evendo them without using a calculator.Q2LE04: It was very, very interesting because I was watchingthe Ndebele drawings, they used transformations and they aregood.Q2LE03: I feel very good when they teach a lesson showing mehow it works.R: Do you think the way you are thinking about mathematics nowis different from the way you were thinking about mathematicsbefore the project?G1LE03: Yes! Mathematics, I thought was very tough. But nowbecause the project I just see it as a subject which everyoneis doing even outside school.R: Do you think that the way you have been asking questionsabout mathematics before the project is different from the wayyou can ask questions now?GILE03: It has changed. Before I used to think of questionslike … Does this work? How does transformation work? Where dowe use transformations and so forth… Yeah, but now I won’tthink or ask myself such questions. I will start thinking orlooking for places where such things can be applied in ourcultures. Yeah I now know our cultures use such mathematics.

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CommentAlthough using the identity-as-narrative construct may soundvalid, when it comes to performance, there is a need fortriangulation. Because of this need, learners’ narratives werecross-checked against observation data, i.e. checking howlearners participated in the culturally-based lessons. Lessonvideos were replayed. This then entailed rigorous validationof data procedures. Learners valued sharing of ideas with their teachers andclassmates (J1LD01). Being part of a group and working ascollective enabled learners to share their indigenousmathematical knowledge. This sharing of knowledge stated inthe narratives was observed during lessons when discussingcultural incidences where number patterns can be observed(Venda clothes, Zulu baskets and different cultural dances).The mathematics classroom can also be organised to encouragediscussion, sharing and collaboration (Boaler, 2000). Groupactivities were satisfactorily and actively done. The factthat the learners liked the sharing of ideas, this activeparticipation included not only thoughts and actions but alsomembership within a mathematics classroom community. Learningmathematics involves the development of each learner’s identity asa member of the classroom community (Wenger, 1998). Learnerswere observed being central to the community of practice.Through the observed warm relationships and valued experienceswith their peers and teachers, it can be concluded that theselearners came to know who they are relative to mathematics.Due to the non-threatening and supportive environment,learners could actively engage in the activities at levelsthat met their understanding. This ethos has been documentedin Boaler’s studies (Boaler, 2002a, 2002b) as being one thatenables learners to participate without threat and hence openup opportunities for participation and learning. Learners also valued the unique experiences of using culturalcontexts in teaching and learning (J3LE12; Q2LE04). Theseexperiences were unique in the sense that mathematicalknowledge was constructed out of culturally-based activities.“I know how to dance and I now know how to draw a numberpattern from a dancing style” (J1LD02). This contention showsthe learner’s capability to construct mathematical knowledge

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from a culturally-based activity –a dance, sending a messagethat culturally relevant pedagogy can provide a worth-whilelearning experience (see Q2LE17; Q2LE29). The idea ofinvolving learners in practical activities was positivelyreceived by learners (Q2LE03) who seem to like thevisualisation involved. The same statement by Q2LE03 wasobserved by Teacher A during lessons and reiterated in thelast interview: “Personally I have realised learners understand more easilywhen the lesson is practical. Yeah… when learners are involved in a lesson theyparticipate with interest” [emphasis placed] (TR A, Final interview,16/03/2012). The practical work provided a range ofaffordances and issues related to mathematics and mathematicslearning – learners’ mathematical identity. Some learners arenow thinking of searching for mathematical knowledge fromcultural activities (GILE03). Therefore identities get formedin practice (Lave & Wenger, 1991). Vignette 4: Recognition as an Identity dimensionSelf-recognition

GILE03: I see myself as a whiz-kid in mathematics (Group-interview, 05/03/2012) GILE01: I see myself as a whiz-kid too because I can nowunderstand mathematics. GILE02: Me, I can see myself as a champion because I understandeverything we learnt in the project.GILB02: Mathematics is now simple to me.GILB01: I can now understand mathematics better whenmathematics examples from our South African cultures are used.GILB03: I think much has changed because I used to see culturalactivities as activities which just ended at the village butnow … now I can use the activities to understand themathematics we learn at school, to simplify my work.GILE02: Yes, before the project I used to consider mathematics as tough. Even when we were given some work to write, I used not to write, but now I am so happy. I did not like mathematics; I hated it so much but now … GILB04: In future we will try to use the same method we used inthe project to learn mathematics, thinking about how the samemathematics is being used in our cultures.

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Comment

Most learners in the study described themselves as capablemathematics learners; they began to imagine themselves asfitting into this community of practice. Imagination involves“the creative process of producing new ‘images’ and ofgenerating new relations through time and space that becomeconstitutive of self” (Wenger, 1998, p. 177). One hundred andsixty-four out of 171 (95%) learners positively acknowledgedculturally relevant pedagogy as enabling them to understandmathematical topics taught in the lessons better. Somelearners emphasised that their engagement in culturallyrelevant lessons will significantly influence their futuremathematics learning (GILE02; GILB04)). However, this maydepend on the nature of the mathematical concepts, providingpossible connections to cultural knowledge. Culturallyrelevant lessons also afforded learners with the opportunityto see themselves as mathematics learners even when they areaway from the classroom (GILB03). Learning is being viewed asa trajectory. As trajectories, our identities incorporate thepast and the future in the very process of negotiating thepresent. “Community of practice is a living context that cangive newcomers access to competence and also invites personalexperience of engagement by which to incorporate thatcompetence into an identity of participation” (Wenger, 1998,p.214).

Recognition by others

TR B: The funny thing is that the learners were able tomake their own explanations from what they saw, theycould visualise everything and could deduct theirexplanation from that. (Final interview, 15/03/2012)

TR A: Learners were involved because they activelyparticipated in the discussions and at least theyunderstood the lessons easily. (Final interview,16/03/2012)

TR A: I realised that practically, it is the way to goand learners can do better when their cultures areinvolved. (Final interview, 16/03/2012)

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TR B: In my lessons only three or four learnersparticipate, but in these lessons almost all learnersparticipate. In groups I could see they were sharingideas. I also observed that almost all the learnerssubmitted the given tasks unlike in my previous lessons.Most learners do not write home work. (Notes ofreflective meetings, 21/2/2012)

CommentThis recognition of learners by their teachers, as becomingcapable mathematics leaners due to their engagement in theculturally relevant lessons happened repeatedly especiallywhen reflecting on the lessons. According to the observationby TR B, learners were involved in mathematical thinkingbecause they could come up with their own explanations.Learners also made an effort to complete and submit giventasks. Shannon, (2007) posits that a realistic context willfacilitate student success by intrinsically motivatingstudents and thus increasing the likelihood that they willmake a serious effort to complete given problems. This isanother indication of successful participation in themathematics classroom community. All learners can becomemathematics learners, identifying themselves and beingrecognised by others as capable of doing mathematics(Anderson, 2007, p.13).

Learning from learners’ narrative storiesThe majority of the learners in the study expressed theirexperiences in culturally-based lessons in terms ofcompetence, performances and recognition. Their engagement inthe lessons led to the definitions of their actions ascompetent members of the mathematics classroom community.Contrary to their initial thinking that the way teachers teachcontribute to their non-participation in the mathematicsclassroom community, learners perceived culturally relevantpedagogy as an enabling pedagogy which can provide access tomathematics. They valued their experiences in the culturally-

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relevant lessons as unique, where mathematical “knowledge waspulled out” of culturally-based activities (Ladson-Billings,1995, p.479) and the construction of knowledge was part of thelearning experiences. They recognised themselves and wererecognised by their teachers as now capable mathematicslearners. What then can we learn from their stories?

The dimensions of identity reflected in this paper can be usedto understand how learners label themselves as members of themathematics community in relation to teaching and learningstrategies. Thus these findings interrogate a strategy fordrawing learners into mathematics, stimulating theirinterests, enthusiasm, competence and confidence. Learners’experiences may not necessarily reflect only one of the threedimensions of identity described in this paper. All the threeor two dimensions can be intersected, therefore leading to aneven stronger mathematical identity. One learner can labelhimself/herself as good in mathematics due to the experiencedcompetence through participating in some unique experiences.Teachers can provide learners with opportunities to seethemselves as mathematics learners away from the classroom(Anderson, 2007). The various images learners have forthemselves and of mathematics extending outside the classroom- in the past, present, or future may change over time(learning trajectories) and this change may be influenced bythe espoused pedagogies. ConclusionThrough consistent and sustained efforts by mathematicsteachers to develop positive identities in their learners, weargue that more learners can improve their identities asmathematics learners. As pointed throughout this paper,identities are socially developed in relationships withothers, including teachers and peers in the learningcommunity. We designed a mathematics identity model whichenabled us to look at what happened within culturally relevantmathematics lessons. This model highlighted competence,performance and recognition as interrelated dimensions fordescribing learning as an identity (Wenger, 1998). The use ofculturally relevant pedagogy positively impacted on learners’

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mathematical identities since the learning trajectoriesreflected in the learners’ narratives show that learners weredrawn more centrally into their community of practice,recognising themselves as competent mathematics learnersbasing on the two topics taught in the study and theirperceived future anticipations.

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